Subtracting Fractions with Unlike Denominators

When we subtract fractions with like denominators, we ignore the denominator, subtract the numerators, and put the result over the denominator. What if the fractions don't have the same denominator though? In that case, we look for equivalent fractions that do have a common denominator by finding the LCM (least common multiple) between the two denominators.

In this article, we'll look at sample problems to better understand how to subtract fractions with unlike denominators. Let's get started!

Proper procedure for subtracting fractions with unlike denominators

Let's say we want to find the difference between $\frac{6}{7}$ and $\frac{2}{3}$ . We can write that mathematically like this:

$\frac{6}{7} - \frac{2}{3}$

Or, we can express it with pictures:

Either way, we have to find the LCM of the denominators: 7 and 3. The LCM between them is 21, so we need to convert both fractions into something equivalent with 21 as the dominator. We can do that by multiplying the numerator and denominator of $\frac{6}{7}$ by 3 and $\frac{2}{3}$ by 7, giving us the following fractions:

$\frac{6 \times 3}{7 \times 3} - \frac{2 \times 7}{3 \times 7}$

$\frac{18}{21} - \frac{14}{21}$

We have like denominators now, so we can subtract normally. $18 - 14 = 4$ , and we put that over our shared denominator of 21 for an answer of $\frac{4}{21}$ . We would convert this answer to the simplest form if we could, but $\frac{4}{21}$ is already in the simplest form. Here is a graphic illustrating what we just did:

If you aren't great at finding the LCM between the two denominators, you can always multiply them together. Sometimes, the product will be the LCM such as in the example above $(7 \times 3 = 21)$ . Sometimes, you won't get the least common multiple but you will get a multiple. For instance, let's say the two denominators are 6 and 9. The LCM is 18, but their product is 54. 54 isn't the least common multiple, but we can still use 54 to get equivalent fractions and reduce our answer when we're done subtracting. Working with the LCM doesn't guarantee your answer will be in simplest form anyway, so don't spend too much time trying to find the LCM.

Practice questions on subtracting fractions with unlike denominators

a. Subtract and simplify: $\frac{3}{4} - \frac{2}{5}$

When subtracting fractions with unlike denominators, the first step is always getting the denominators to be the same. The LCM of 4 and 5 is 20, so we multiply $\frac{3}{4}$ by 5 and $\frac{2}{5}$ by 4 to get equivalent fractions:

$\frac{3 \times 5}{4 \times 5} - \frac{2 \times 4}{5 \times 4}$

$\frac{15}{20} - \frac{8}{20}$

From here, we can subtract normally. $15 - 8 = 7$ . Next, put the 7 over the shared denominator of 20 for a final answer of 7/20.

b. Subtract and simplify: $\frac{3}{5} - \frac{3}{10}$

When subtracting fractions with unlike denominators, the first step is getting the denominators to be the same. The LCM of 5 and 10 is 10, so we multiply 3/5 by 2 and leave 3/10 alone since it already has the denominator we're looking for. That gives us:

$\frac{3 \times 2}{5 \times 2} - \frac{3}{10}$

$\frac{6}{10} - \frac{3}{10}$

Now that we have fractions with like denominators, we can take the difference of the numerators (6 - 3 = 3) and put it over the shared denominator of 10 for an answer of $\frac{3}{10}$ .

c. Subtract and simplify: $\frac{7}{8} - \frac{3}{4}$

We're subtracting fractions with unlike denominators, so the first step is finding equivalent fractions with a shared denominator. The LCM is 8, but let's pretend we don't see that and multiply the denominators together to get a shared multiple. $8 \times 4 = 32$ . Now, we want equivalent fractions with a denominator of 32. That gives us:

$\frac{7 \times 4}{8 \times 4} - \frac{3 \times 8}{4 \times 8}$

$\frac{28}{32} - \frac{24}{32}$

$\frac{4}{32}$

Our answer isn't in simplest form because we didn't use the LCM, but that's okay. 4 goes into 32: 8 times, so we can simplify our answer to $\frac{1}{8}$ . That's what we would get if we used the LCM!

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